Find the sum of first 22 terms of an A.P. in which d = 7 and `22^(nd)` term is 149.

Options

A. 1661
B. 1601
C. None of these
D. 1194

Answer

To solve the problem, we need to find the sum of the first 22 terms of an Arithmetic Progression (A.P.) where the common difference \( d = 7 \) and the 22nd term \( A_{22} = 149 \).

### Step-by-Step Solution:

1. **Understand the formula for the nth term of an A.P.**:
The nth term of an A.P. can be expressed as:
\[ A_n = A + (n-1) \cdot d \]
where \( A \) is the first term, \( d \) is the common difference, and \( n \) is the term number.

2. **Substitute the known values into the formula for the 22nd term**:
Given that \( n = 22 \), \( d = 7 \), and \( A_{22} = 149 \), we can write:
\[ A_{22} = A + (22 - 1) \cdot 7 \]
This simplifies to:
\[ 149 = A + 21 \cdot 7 \]

3. **Calculate \( 21 \cdot 7 \)**:
\[ 21 \cdot 7 = 147 \]
Thus, the equation becomes:
\[ 149 = A + 147 \]

4. **Solve for \( A \)**:
Rearranging the equation gives:
\[ A = 149 - 147 = 2 \]

5. **Now, find the sum of the first 22 terms \( S_{22} \)**:
The formula for the sum of the first \( n \) terms of an A.P. is:
\[ S_n = \frac{n}{2} \cdot (2A + (n-1) \cdot d) \]
Substituting \( n = 22 \), \( A = 2 \), and \( d = 7 \):
\[ S_{22} = \frac{22}{2} \cdot (2 \cdot 2 + (22 - 1) \cdot 7) \]

6. **Calculate \( 2 \cdot 2 \)**:
\[ 2 \cdot 2 = 4 \]

7. **Calculate \( (22 - 1) \cdot 7 \)**:
\[ 21 \cdot 7 = 147 \]

8. **Combine the results**:
\[ S_{22} = 11 \cdot (4 + 147) = 11 \cdot 151 \]

9. **Calculate \( 11 \cdot 151 \)**:
\[ S_{22} = 1661 \]

### Final Answer:
The sum of the first 22 terms of the A.P. is \( S_{22} = 1661 \).